# Absolute Value Equations

Equations with a variable or variables within absolute value bars are known as absolute value equations .

When solving equations that involve absolute values, there are two cases to consider.

Case 1: The expression inside the absolute value bars is positive.

Case 2: The expression inside the absolute value bars is negative.

For example, consider the expression $|\text{\hspace{0.17em}}4x+2\text{\hspace{0.17em}}|=8$ .

For this to be true, either

$4x+2=8$

OR

$4x+2=-8$ .

You need to solve both equations. In this case, the solution to the first one is $x=4$ , and the the solution to the second one is $x=-5$ . So there are two solutions , $x=4$ and $x=-5$ .

It's also possible for an absolute value equation to have one solution :

$|\text{\hspace{0.17em}}x+3\text{\hspace{0.17em}}|=0$ has the single solution $x=-3$

or no solutions:

$|\text{\hspace{0.17em}}5x+1\text{\hspace{0.17em}}|=-6$

(The absolute value of any expression is positive, so there is no value of $x$ for which this is true.)